was introduced for the first time in [1]}. The reconstructions of the smooth atlas on the manifold and the metric tensor in these coordinates was originally considered in [2]}. In contrast to the paper at hand these earlier results do not need any extra assumption for the geometry, but have a complete data in the sense that the measurement area \(\Gamma \)  is the whole boundary. Their counterpart of our Theorem REF , is [2]}, which says that the boundary distance function \(\hat{r}_p\)  is smooth near its minimizers (the set of closest boundary points to the source point \(p\in M\) ). This lemma is a key of their proof. However, the same technique is not available to us as it requires the access to the whole boundary.
